ガロア第一論文と乗数イデアル他関連資料スレ6

[過去ﾛｸﾞ] ガロア第一論文と乗数イデアル他関連資料スレ6 (1002ﾚｽ)
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157: １３２人目の素数さん [] 2024/01/25(木) 10:37:02.23 ID:zxKJrX2I

>>149 訂正と補足

＜訂正＞
・それが、ミルナーのh-cobordismにつながり
　　↓
・それが、Smaleのh-cobordismにつながり

(参考)
https://en.wikipedia.org/wiki/H-cobordism
h-cobordism
In geometric topology and differential topology, an (n + 1)-dimensional cobordism W between n-dimensional manifolds M and N is an h-cobordism (the h stands for homotopy equivalence) if the inclusion maps
M → W and N → W
are homotopy equivalences.

The h-cobordism theorem gives sufficient conditions for an h-cobordism to be trivial, i.e., to be C-isomorphic to the cylinder M × [0, 1]. Here C refers to any of the categories of smooth, piecewise linear, or topological manifolds.

The theorem was first proved by Stephen Smale for which he received the Fields Medal and is a fundamental result in the theory of high-dimensional manifolds. For a start, it almost immediately proves the generalized Poincaré conjecture.

Background
Before Smale proved this theorem, mathematicians became stuck while trying to understand manifolds of dimension 3 or 4, and assumed that the higher-dimensional cases were even harder. The h-cobordism theorem showed that (simply connected) manifolds of dimension at least 5 are much easier than those of dimension 3 or 4. The proof of the theorem depends on the "Whitney trick" of Hassler Whitney, which geometrically untangles homologically-tangled spheres of complementary dimension in a manifold of dimension >4. An informal reason why manifolds of dimension 3 or 4 are unusually hard is that the trick fails to work in lower dimensions, which have no room for entanglement.
(引用終り)

＜補足＞
　>>149の福田拓生先生が、ルネ・トムから直接聞いた話の素直な解釈は
　H.Cartan：岡論文を読め
　　↓
　岡論文：上空移行 次元を上げよ
　　↓
　ルネ・トム：岡先生ありがとう、+１次元のコボルディズムが閃いた
　　↓
　ルネ・トム：「日本で岡先生に会えたときには感激した」と懐かしそうに言われた

ということではないでしょうか

http://rio2016.5ch.net/test/read.cgi/math/1704672583/157

162: １３２人目の素数さん [] 2024/01/25(木) 12:21:00.01 ID:zxKJrX2I

>>161
そうですね

岡論文：上空移行 次元を上げよ
が
ルネ・トム：+１次元のコボルディズムのヒントになり

またそれが、Smaleのh-cobordismによる 高次元ポアンカレ予想解決になった>>157
別に、John MilnorのSurgery theory（手術理論）が発展しました
Milnorさんもフィールズ賞です
そして、（3次元）ポアンカレ予想にも、Surgery theory（手術理論）が使われた（これもフィールズ賞）

”岡論文：上空移行”は、偉大ですね

(参考)
https://en.wikipedia.org/wiki/Surgery_theory
Surgery theory
In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by John Milnor (1961).
A relatively easy argument using Morse theory shows that a manifold can be obtained from another one by a sequence of spherical modifications if and only if those two belong to the same cobordism class.[1]
Attaching handles and cobordisms
A surgery on M not only produces a new manifold M′, but also a cobordism W between M and M′. The trace of the surgery is the cobordism (W; M, M′), with
略

https://en.wikipedia.org/wiki/John_Milnor
John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook University and the only mathematician to have won the Fields Medal, the Wolf Prize, the Abel Prize and all three Steele prizes.

https://ja.wikipedia.org/wiki/%E3%83%9D%E3%82%A2%E3%83%B3%E3%82%AB%E3%83%AC%E4%BA%88%E6%83%B3
（3次元）ポアンカレ予想
幾何化予想とペレルマン
ペレルマンは、特異点が発生する3次元多様体に対して、3次元手術つきリッチフロー (Ricci flow with surgery) を適用することによって幾何化予想を解決した[14]。手術とは、有限時間で生成する特異点の直前でシリンダー状の部分の切り口 S2 に沿って球面状のキャップをかぶせてそこに標準解と呼ばれるものを貼ることである[2][14][15]。ペレルマンは、この手術を特異点が生成する時空の点に限りなく近づける極限をとることにより、3次元リッチフローが有限時間での特異点を超えて標準的に延長することを証明した[2][14][16]。

http://rio2016.5ch.net/test/read.cgi/math/1704672583/162

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